Finally I managed to update my crappy orthographic drawer. Okay, so this only has fake shadowing (the same technique as the superposition of the earlier broccoli), but here's another 'taster' for what we can expect this thing to look like. I'm looking forward to adding proper shadowing soon. Enjoy, and click here for full size.

Sorry for the above spamming, but this is actually pretty exciting. It's amazing what a simple formula can do. The render below consists of a zoom in the bottom right corner of the earlier 'golden' mandelbrot, but this time in full shadowed mode. It uses 7 iterations. The render took around 16-24 hours on a single core, and consists of around 2000x2000x20000 voxels. For each pixel, around 200 rays were shot in spherical directions to obtain the shadows.

but, beside of spamming, i also want to use the 'new' developed formulas, as far as i know yet you use actually 2 variants for using a true 3 Dimensional Vector and you have extended the 2 dimensional complex calculations to a 3 dimensional method which differs from the normally used quaternionic 4d method which compliescertain mathematical rules ( for complex numbers it was : i*i=-1 )....

i think we should create a (pdf,latex?) document describing the modifications, and list the formulas, as it was done with the fractal flame algorithm ... this would makefor a nice coproduction on fractalforums.com i can start on creating a rough outline for the document, and start collecting the formulas you have developed ... and tryto explain why the modifications took place ...

the document should contain the following parts:

prerequisite: complex numbers description extending to "real" 3d ( not using 3 parts of a 4d vector ) A Chapter About Coloring/inside methods, because outlining the "inner" part of the fractal ( the points which do not diverge ) is just one coloring method out of many ( HPDZ has a nice document describing that http://www.hpdz.net/TechInfo_Colorizing.htm )

That all sounds good. Colouring the thing will produce even more incredibleness. I will stop short of calling my above renders a 'glimpse', but full global illumination and (most dramatically) perspective will make this 100x better yet still!! When stuff's in 3D, it makes all the difference. In the gateaux pic, imagine standing on one of the 'balconies' and looking across and down the surface on the mountain, with sumptious detail to either side. It will kill the CPU as always, but wow, to even think of this is too awesome.

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and you have extended the 2 dimensional complex calculations to a 3 dimensional method which differs from the normally used quaternionic 4d method which complies certain mathematical rules ( for complex numbers it was : i*i=-1 )....

Yup - pretty much. It's the same formula I printed back here, except where the squaring changes to higher powers instead. Hopefully these results justify my obsession with all of this to begin with

Of course, me and Paul can't take all the credit for the concept. To my surprise, it turns out that Rudy Rucker also thought of the same formula around 20 years ago (and didn't realise its potential), and for all I know, others have thought of it too. Of course, limited computer speed would hinder research back then, and Paul's addition of highers powers for some utterly strange reason, actually produces all this detail.

Of course, what we have here is much closer, but it's still not quite the holy grail I'm after. I could definitely be fooled though! Though I do have my suspicions the real thing will look yet better still.

Here is a formula for the square root based on Twinbee's formula for squaring a 3D hypercomplex number ("triplex"). By definition, the square root must satisfy sqrt({x,y,z})▓ = {x,y,z}. As it turns out, 4 roots can be found that satisfy this equation. Using these roots, one can calculate inverse Julia set fractals. This method is very fast, but some regions are faint because they attract much slower. For example, 2 billion points can be calculated in a matter of seconds, although rendering can take longer. Here are some pictures:

If you look closely there is a image of the formula underneath my previous post. Unfortuntely, Fractal Forums continues to insist on shrinking my images into tiny thumbnails so it's easy to overlook the image if you're not looking carefully for it.

If you look closely there is a image of the formula underneath my previous post. Unfortuntely, Fractal Forums continues to insist on shrinking my images into tiny thumbnails so it's easy to overlook the image if you're not looking carefully for it.

I see it now - was it that big originally ? If so I can't believe I missed it !

Edit: Thanks for posting the C version

Edit2: I know why I missed the gif formula, I was reading your post in "recent posts" and the images don't get shown there.

David, I cannot find a division formula for your multiplication formula because it is expressed in trigonometric form. I was able to find a square root formula using the "simplified" non-trigonometric form of the squaring equation, but I can't seem to obtain a similar non-trigonometric form of your multiplication formula.

Just out of curiosity, what were you hoping to do with this division formula?

David, I cannot find a division formula for your multiplication formula because it is expressed in trigonometric form. I was able to find a square root formula using the "simplified" non-trigonometric form of the squaring equation, but I can't seem to obtain a similar non-trigonometric form of your multiplication formula.

Just out of curiosity, what were you hoping to do with this division formula?

Ermmm, "true 3D" formulas using division - such as different versions of the Newton or Nova.